Geometry of Kähler Metrics and Foliations by Holomorphic Discs

2005/07/07 by X. X. Chen, Xijuan Chen, Gang Tian +3 · 5 citations
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.math/0507148

93 pages

arxiv created 2005/07/07 · openalex publication_date 2005/07/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and their relations to homogeneous complex Monge-Ampere equations. As applications, we will prove the uniqueness of extremal Kähler metrics and give an necessary condition for existence of extremal Kähler metrics. Further applications will be discussed in our forthcoming papers.

Cited by

• Convexity of the K-energy and Uniqueness of Extremal metrics -- An Expository Introduction
• On the effective YTD conjecture
• Test configurations and Geodesic rays
• A stronger concept of K-stability
• Lines in the space of Kähler metrics

Related

• Geometry of Kaehler metrics and holomorphic foliation by discs
• A uniform L^∞ estimate for complex Monge-Ampere equations
• Asymptotic Behavior of Homogeneous Complex Monge-Ampere Equations on ALE Kahler manifolds
• Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics
• Homogeneous Monge-Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry